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. SSSEC(The Intuition for Squaring)

The notion  of  "Intuition" is  so strange  that I  shall take  every
opportunity to  try to explicate it. This  subsection is a digression
to that purpose,  and may therefore be  skipped forever, returned  to
later, or read right now.

Consider  what  happens  when   the  task  ⊗6"Fill-in  intuitions  of
TIMES-Itself"⊗*  gets  chosen.   Rippling  upward, we  find  that the
intuition for TIMES is a  rectangle (in the plane) built out  of unit
squares, where  the length of  each side of  the rectangle represents
the two arguments to TIMES, and the area of the rectangle (i.e.,  the
rectangle itself)  represents the value  of TIMES.   For the  current
operation, this translates to a rectange whose sides are all of equal
length: i.e., a  square. If AM  is given  this notion explicitly,  it
will actually suggest renaming the TIMES-Itself operation "Square".

Notice  the   nice  intuitive   image  that   this  will   cause  for
square-rooting,  since a number n  is represented as a  pile of n 1x1
squares. To take  the square-root of a  number, we must arrange  that
pile into one large square, and then see how long one side is.

This also suggests several trivial relationships to verify. Since the
number 1 is intuited as a  1x1 square, it seems intuitively clear  to
AM that  Times-Itself(1) will be  equal to 1.   This is  then checked
"officially"  (using the  definition  of Times-Itself)  and verified.
Similar considerations suggest that Times-Itself(0) will equal 0, and
that in general Times-Itself(x) is much larger than x.

Although AM never noticed it, this intuition is just the right one to
notice  that to get from  n⊗A2⊗* to n+1⊗A2⊗*, it  is necessary to add
2n+1 unit squares: i.e., that n⊗A2⊗*  + 2n + 1 = n+1⊗A2⊗*. This  is a
far  from  trivial result  to  notice, when  one  doesn't know  about
algebra. In fact, AM couldn't verify such a general conjecture, since
it knows neither the concept of  mathematical induction, nor anything
about proof in general.